Note that sampling \(S^{\prime } \subseteq I_p\)  is equivalent to making a binary decision for each \(v \in I_p\) ; whether to put \(v\)  into \(S^{\prime }\)  or not. In this interpretation, we can almost directly import the tools for CSPs (with binary alphabets). For sake of completeness, we show how the framework of Raghavendra and Tan [1]} used for Max-CSPs with cardinality constraints can be used for our purpose. Similar techniques also have been used for non-constrained CSPs and graph partitioning problems [2]}, [3]}.
